Reconstructing the topology of clones

Reconstructing the topology on monoids and polymorphism clones of the rationals

We show how to reconstruct the topology on the monoid of endomorphisms of the rational numbers under the strict or reflexive order relation, and the polymorphism clone of the rational numbers under the reflexive relation. In addition we show how automatic homeomorphicity results can be lifted to polymorphism clones generated by monoids.

Reconstructing the topology on monoids and polymorphism clones of reducts of the rationals

We extend results from an earlier paper giving reconstruction results for the endomorphism monoid of the rational numbers under the strict and reflexive relations to the first order reducts of the rationals and the corresponding polymorphism clones. We also give some similar results about the coloured rationals.

Polymorphism clones of homogeneous structures (Universal homogeneous polymorphisms and automatic homeomorphicity)

Every clone of functions comes naturally equipped with a topology---the topology of pointwise convergence. A clone $\mathfrak{C}$ is said to have automatic homeomorphicity with respect to a class $\mathcal{C}$ of clones, if every clone-isomorphism of $\mathfrak{C}$ to a member of $\mathcal{C}$ is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. To this end we introduce and utilize universal homogeneous polymorphisms. Next to two gen…

关于自同态幺半群和多态克隆上拓扑重构的指南

A guide to topological reconstruction on endomorphism monoids and polymorphism clones

综述自同态幺半群和多态克隆上拓扑重构研究，介绍结构对称空间的代数与拓扑结构，回顾自同构群相关问题研究历程，阐述对上述类似问题的研究现状，还完善并扩展了一些现有结果。

Reconstructing the topology of the elementary self-embedding monoids of countable saturated structures

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever $\mathbf{A}$ is…

Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do wi…